3113:
3176:
3491:
3170:
3617:
Conversely, every formally real field can be equipped with a compatible total order, that will turn it into an ordered field. (This order need not be uniquely determined.) The proof uses
3227:
2860:
2799:
2728:
3839:
2148:
3055:
2976:
2591:
2227:
2345:
2311:
2280:
2179:
1875:
475:
2543:
2096:
223:
1071:
2948:
2904:
2249:
2201:
1826:
1797:
1029:
781:
385:
2657:
2624:
1701:
990:
315:
3404:
2491:
2021:
2394:
1924:
1763:
1736:
1648:
900:
2044:
1558:
954:
634:
582:
556:
530:
440:
414:
344:
3011:
712:
686:
660:
608:
504:
1180:
246:
157:
2452:
2423:
1982:
1953:
1275:
874:
1581:
1530:
1459:
1414:
1369:
1322:
1207:
1153:
1119:
1096:
923:
848:
1668:
1621:
1601:
1507:
1487:
1436:
1391:
1342:
1295:
1249:
1229:
825:
805:
752:
266:
3398:
0. (Since 1 > 0, then 1 + 1 > 0, and 1 + 1 + 1 > 0, etc., and no finite sum of ones can equal zero.) In particular, finite fields cannot be ordered.
4926:
3112:
4909:
4439:
4275:
4964:
4237:
3504:(as for any other field of characteristic 0), and the order on this rational subfield is the same as the order of the rationals themselves.
4756:
167:
in 1926, which axiomatizes the subcollection of nonnegative elements. Although the latter is higher-order, viewing positive cones as
4892:
4751:
4207:
4174:
4959:
4746:
3947:
4382:
4199:
61:
of an ordered field is also an ordered field in the inherited order. Every ordered field contains an ordered subfield that is
4464:
3711:
3119:
4783:
4703:
4166:
3514:
4377:
4568:
4497:
3496:
Every subfield of an ordered field is also an ordered field (inheriting the induced ordering). The smallest subfield is
4471:
4459:
4422:
4397:
4372:
4326:
4295:
3507:
If every element of an ordered field lies between two elements of its rational subfield, then the field is said to be
4402:
4392:
116:
3175:
4768:
4268:
3858:
3583:
3104:, but otherwise obey the axioms of an ordered field. Every ordered field can be embedded into the surreal numbers.
3182:
2804:
2733:
2662:
4741:
4407:
3576:
112:
3778:
2101:
4673:
4300:
3628:
3395:
4969:
4921:
4904:
3964:
4833:
4449:
4044:
3958:
3923:
4811:
4646:
4637:
4506:
4341:
4305:
4261:
3595:
3058:
2024:
46:
of its elements that is compatible with the field operations. Basic examples of ordered fields are the
4387:
3023:
2953:
2548:
2206:
4899:
4858:
4848:
4838:
4583:
4546:
4536:
4516:
4501:
3952:
3509:
2319:
2285:
2254:
2153:
1849:
445:
2496:
2049:
189:
4826:
4737:
4683:
4642:
4632:
4521:
4454:
4417:
3611:
3591:
3582:) over an ordered field exhibit some special properties and have some specific structures, namely:
3486:{\displaystyle \textstyle \sum _{k=1}^{n}a_{k}^{2}=0\;\Longrightarrow \;\forall k\;\colon a_{k}=0.}
1034:
184:
120:
39:
2909:
2865:
2232:
2184:
1809:
1780:
995:
760:
349:
4865:
4718:
4627:
4617:
4558:
4476:
3097:
2629:
2596:
2348:
1878:
1673:
959:
279:
4938:
4778:
4412:
4037:
2461:
1991:
103:
of an ordered field was abstracted gradually from the real numbers, by mathematicians including
2353:
1883:
1741:
1706:
1626:
4875:
4853:
4713:
4698:
4678:
4481:
4233:
4203:
4170:
3760:
3752:
3719:
3647:
3557:
3526:
3101:
3074:
3014:
1840:
1836:
879:
136:
70:
2029:
1537:
930:
613:
561:
535:
509:
419:
393:
323:
4688:
4541:
4243:
4213:
4180:
3756:
3086:
3080:
2981:
691:
665:
639:
587:
483:
160:
74:
3656:
contains a square root of −7, thus 1 + 1 + 1 + 2 +
1158:
231:
142:
89:
4870:
4653:
4531:
4526:
4511:
4336:
4321:
4247:
4217:
4184:
3854:
3710:
arising from the total order ≤, then the axioms guarantee that the operations + and × are
3587:
3530:
3501:
3093:
2428:
2399:
1958:
1929:
1800:
66:
58:
47:
4427:
3618:
1254:
853:
1563:
1512:
1441:
1396:
1351:
1304:
1189:
1135:
1101:
1078:
905:
830:
4788:
4773:
4763:
4622:
4600:
4578:
4159:
3970:
3707:
3640:
3632:
3534:
1653:
1606:
1586:
1492:
1472:
1421:
1376:
1327:
1280:
1234:
1214:
810:
790:
737:
251:
108:
100:
82:
78:
4953:
4843:
4821:
4693:
4563:
4551:
4356:
4191:
4154:
3929:
3926: – Group with translationally invariant total order; i.e. if a ≤ b, then ca ≤ cb
3850:
3846:
3635:
also cannot be turned into an ordered field, as −1 is a square of the imaginary unit
3518:
3391:. In particular, since 1=1, it follows that 0 ≤ 1. Since 0 ≠ 1, we conclude 0 < 1.
104:
131:
There are two equivalent common definitions of an ordered field. The definition of
4708:
4590:
4573:
4491:
4331:
4284:
3973: – Algebraic concept in measure theory, also referred to as an algebra of sets
3938:
3624:
3572:
1534:
Given a field ordering ≤ as in the first definition, the set of elements such that
93:
51:
4914:
4607:
4486:
4351:
3979:
3522:
3497:
3069:
1829:
226:
62:
43:
31:
17:
4882:
4816:
4657:
4225:
2455:
1985:
4933:
4806:
4612:
3747:. Each order can be regarded as a multiplicative group homomorphism from
4728:
4595:
4346:
3914:
is a totally real field in which the set of sums of squares forms a fan.
3842:
1183:
171:
prepositive cones provides a larger context in which field orderings are
77:
are necessarily non-negative in an ordered field. This implies that the
3602:, which can be generalized to vector spaces over other ordered fields.
1843:, becomes an ordered field by restricting the ordering to the subfield;
3013:
is greater than any constant polynomial and the ordered field is not
755:
3174:
3111:
1438:
are precisely the intersections of families of positive cones on
4165:, CBMS Regional Conference Series in Mathematics, vol. 52,
1623:
as in the second definition, one can associate a total ordering
1489:
be a field. There is a bijection between the field orderings of
4257:
4253:
1832:
with its standard ordering (which is also its only ordering);
1803:
with its standard ordering (which is also its only ordering);
3982: – Partially ordered vector space, ordered as a lattice
3401:
Every non-trivial sum of squares is nonzero. Equivalently:
3676: > 2) contains a square root of 1 −
3310:
One can "multiply inequalities with positive elements": if
3631:
cannot be turned into ordered fields, as shown above. The
3614:, i.e., 0 cannot be written as a sum of nonzero squares.
4093:, but are < 0, so that these roots cannot be in
276:
if the order satisfies the following properties for all
3975:
Pages displaying short descriptions of redirect targets
3934:
Pages displaying short descriptions of redirect targets
3564:. This property implies that the field is Archimedean.
3408:
3333:"Multiplying with negatives flips an inequality": if
3165:{\displaystyle a>0\land x<y\Rightarrow ax<ay}
2282:. In this fashion, we get many different orderings of
3781:
3533:
real numbers with elements greater than any standard
3407:
3185:
3122:
3026:
2984:
2956:
2912:
2868:
2807:
2801:, respectively. Equivalently: for rational functions
2736:
2665:
2632:
2599:
2551:
2499:
2464:
2431:
2402:
2356:
2322:
2288:
2257:
2235:
2209:
2187:
2156:
2104:
2052:
2032:
2023:, can be made into an ordered field by fixing a real
1994:
1961:
1932:
1886:
1852:
1812:
1783:
1744:
1709:
1676:
1656:
1629:
1609:
1589:
1566:
1540:
1515:
1495:
1475:
1444:
1424:
1399:
1379:
1354:
1330:
1307:
1283:
1257:
1237:
1217:
1192:
1161:
1138:
1104:
1081:
1037:
998:
962:
933:
908:
882:
856:
833:
813:
793:
763:
740:
694:
668:
642:
616:
590:
564:
538:
512:
486:
448:
422:
396:
352:
326:
282:
254:
234:
192:
163:. Artin and Schreier gave the definition in terms of
145:
3943:
Pages displaying wikidata descriptions as a fallback
4799:
4727:
4666:
4436:
4365:
4314:
3596:
Real coordinate space#Geometric properties and uses
1835:any subfield of an ordered field, such as the real
4232:(Third ed.), Reading, Mass.: Addison-Wesley,
4158:
3833:
3485:
3221:
3164:
3049:
3005:
2970:
2942:
2898:
2854:
2793:
2722:
2651:
2618:
2585:
2537:
2485:
2446:
2417:
2388:
2339:
2305:
2274:
2243:
2221:
2195:
2173:
2142:
2090:
2038:
2015:
1976:
1947:
1918:
1869:
1820:
1791:
1765:satisfies the properties of the first definition.
1757:
1730:
1695:
1662:
1642:
1615:
1595:
1575:
1552:
1524:
1501:
1481:
1453:
1430:
1408:
1385:
1363:
1336:
1316:
1289:
1269:
1243:
1223:
1201:
1174:
1147:
1113:
1090:
1065:
1023:
984:
948:
917:
894:
868:
842:
819:
799:
775:
746:
706:
680:
654:
628:
602:
576:
550:
524:
498:
469:
434:
408:
379:
338:
309:
260:
240:
217:
151:
3902: − {0} and not containing −1 then
1461:The positive cones are the maximal preorderings.
2493:, can be made into an ordered field by defining
4202:. Vol. 67. American Mathematical Society.
4269:
3932: – Group with a compatible partial order
3845:for the Harrison topology. The product is a
8:
3961: – Ring with a compatible partial order
3828:
3797:
3222:{\displaystyle x<y\Rightarrow a+x<a+y}
2855:{\displaystyle f(x),g(x)\in \mathbb {R} (x)}
2794:{\displaystyle q(x)=q_{m}x^{m}+\dots +q_{0}}
2723:{\displaystyle p(x)=p_{n}x^{n}+\dots +p_{0}}
4196:Introduction to Quadratic Forms over Fields
4038:"Implicit differentiation with microscopes"
3967: – Partially ordered topological space
92:(which is negative in any ordered field).
4927:Positive cone of a partially ordered group
4276:
4262:
4254:
3941: – ring with a compatible total order
3834:{\displaystyle H(a)=\{P\in X_{F}:a\in P\}}
3462:
3455:
3451:
2143:{\displaystyle p(\alpha )/q(\alpha )>0}
81:cannot be ordered since the square of the
73:ordered field is isomorphic to the reals.
27:Algebraic object with an ordered structure
4161:Orderings, valuations and quadratic forms
3955: – Vector space with a partial order
3870:is a closed subset, hence again Boolean.
3810:
3780:
3548:if and only if every non-empty subset of
3529:form a non-Archimedean field, because it
3470:
3439:
3434:
3424:
3413:
3406:
3184:
3121:
3065:is taken to be infinitesimal and positive
3028:
3027:
3025:
2983:
2964:
2963:
2955:
2911:
2867:
2839:
2838:
2806:
2785:
2766:
2756:
2735:
2714:
2695:
2685:
2664:
2637:
2631:
2604:
2598:
2571:
2562:
2556:
2550:
2512:
2498:
2463:
2430:
2401:
2369:
2355:
2324:
2323:
2321:
2290:
2289:
2287:
2259:
2258:
2256:
2237:
2236:
2234:
2208:
2189:
2188:
2186:
2158:
2157:
2155:
2117:
2103:
2065:
2051:
2031:
1993:
1960:
1931:
1899:
1885:
1854:
1853:
1851:
1814:
1813:
1811:
1785:
1784:
1782:
1749:
1743:
1708:
1684:
1675:
1655:
1634:
1628:
1608:
1588:
1565:
1539:
1514:
1494:
1474:
1443:
1423:
1398:
1378:
1353:
1329:
1306:
1282:
1256:
1236:
1216:
1191:
1166:
1160:
1137:
1103:
1080:
1048:
1036:
1009:
997:
967:
961:
932:
907:
881:
855:
832:
812:
792:
762:
739:
693:
667:
641:
615:
589:
563:
537:
511:
485:
447:
421:
395:
351:
325:
281:
253:
233:
211:
191:
144:
4910:Positive cone of an ordered vector space
4022:
4020:
3998:
3996:
2978:. In this ordered field the polynomial
4010:
4008:
3992:
3544:is isomorphic to the real number field
1132:is a field equipped with a preordering
3734:is a topology on the set of orderings
3646:cannot be ordered, since according to
3627:and more generally fields of positive
3598:for discussion of those properties of
54:, both with their standard orderings.
135:appeared first historically and is a
7:
3684: − 1)⋅1 +
3568:Vector spaces over an ordered field
2150:. This is equivalent to embedding
783:that has the following properties:
4437:Properties & Types (
3456:
1583:Conversely, given a positive cone
1465:Equivalence of the two definitions
25:
4893:Positive cone of an ordered field
3592:positively-definite inner product
3050:{\displaystyle \mathbb {R} ((x))}
2971:{\displaystyle t\in \mathbb {R} }
4747:Ordered topological vector space
4072:The squares of the square roots
3948:Ordered topological vector space
2659:are the leading coefficients of
2586:{\displaystyle p_{n}/q_{m}>0}
2229:and restricting the ordering of
2222:{\displaystyle x\mapsto \alpha }
1773:Examples of ordered fields are:
115:. This grew eventually into the
4200:Graduate Studies in Mathematics
3525:form an Archimedean field, but
3275:One can "add inequalities": if
2340:{\displaystyle \mathbb {R} (x)}
2306:{\displaystyle \mathbb {Q} (x)}
2275:{\displaystyle \mathbb {Q} (x)}
2251:to an ordering of the image of
2174:{\displaystyle \mathbb {Q} (x)}
1988:with rational coefficients and
1870:{\displaystyle \mathbb {Q} (x)}
1186:of the multiplicative group of
470:{\displaystyle 0\leq a\cdot b.}
139:axiomatization of the ordering
4036:Bair, Jaques; Henry, Valérie.
3910:is closed under addition). A
3791:
3785:
3452:
3379:Squares are non-negative: 0 ≤
3195:
3144:
3061:with real coefficients, where
3044:
3041:
3035:
3032:
2994:
2988:
2937:
2931:
2922:
2916:
2893:
2887:
2878:
2872:
2849:
2843:
2832:
2826:
2817:
2811:
2746:
2740:
2675:
2669:
2538:{\displaystyle p(x)/q(x)>0}
2526:
2520:
2509:
2503:
2474:
2468:
2441:
2435:
2412:
2406:
2383:
2377:
2366:
2360:
2334:
2328:
2300:
2294:
2269:
2263:
2213:
2168:
2162:
2131:
2125:
2114:
2108:
2091:{\displaystyle p(x)/q(x)>0}
2079:
2073:
2062:
2056:
2004:
1998:
1971:
1965:
1942:
1936:
1913:
1907:
1896:
1890:
1864:
1858:
1393:together with a positive cone
218:{\displaystyle (F,+,\cdot \,)}
212:
193:
1:
4704:Series-parallel partial order
4167:American Mathematical Society
3698:Topology induced by the order
3515:non-Archimedean ordered field
3513:. Otherwise, such field is a
1066:{\displaystyle 1=1^{2}\in P.}
4965:Ordered algebraic structures
4383:Cantor's isomorphism theorem
4104:expansions are not periodic.
3894:is a subgroup of index 2 in
3874:Fans and superordered fields
3108:Properties of ordered fields
2943:{\displaystyle f(t)<g(t)}
2899:{\displaystyle f(x)<g(x)}
2244:{\displaystyle \mathbb {R} }
2196:{\displaystyle \mathbb {R} }
1821:{\displaystyle \mathbb {R} }
1792:{\displaystyle \mathbb {Q} }
1373:An ordered field is a field
1024:{\displaystyle 0=0^{2}\in P}
776:{\displaystyle P\subseteq F}
380:{\displaystyle a+c\leq b+c,}
4423:Szpilrajn extension theorem
4398:Hausdorff maximal principle
4373:Boolean prime ideal theorem
2950:for all sufficiently large
2652:{\displaystyle q_{m}\neq 0}
2619:{\displaystyle p_{n}\neq 0}
2458:with real coefficients and
1696:{\displaystyle x\leq _{P}y}
985:{\displaystyle x^{2}\in P.}
310:{\displaystyle a,b,c\in F:}
4986:
4769:Topological vector lattice
3890:with the property that if
2486:{\displaystyle q(x)\neq 0}
2016:{\displaystyle q(x)\neq 0}
1769:Examples of ordered fields
1509:and the positive cones of
4291:
3906:is an ordering (that is,
3743:of a formally real field
3610:Every ordered field is a
2389:{\displaystyle p(x)/q(x)}
1919:{\displaystyle p(x)/q(x)}
1758:{\displaystyle \leq _{P}}
1731:{\displaystyle y-x\in P.}
1643:{\displaystyle \leq _{P}}
1560:forms a positive cone of
1324:The non-zero elements of
662:, respectively. Elements
4378:Cantor–Bernstein theorem
3751:onto ±1. Giving ±1 the
1211:If in addition, the set
895:{\displaystyle x\cdot y}
4960:Real algebraic geometry
4922:Partially ordered group
4742:Specialization preorder
4122:Lam (1983) pp. 1–2
4097:which means that their
3965:Partially ordered space
3552:with an upper bound in
2039:{\displaystyle \alpha }
1553:{\displaystyle x\geq 0}
949:{\displaystyle x\in F,}
629:{\displaystyle a\leq b}
577:{\displaystyle b\geq a}
551:{\displaystyle a\neq b}
525:{\displaystyle a\leq b}
435:{\displaystyle 0\leq b}
409:{\displaystyle 0\leq a}
339:{\displaystyle a\leq b}
4408:Kruskal's tree theorem
4403:Knaster–Tarski theorem
4393:Dushnik–Miller theorem
3959:Partially ordered ring
3924:Linearly ordered group
3835:
3606:Orderability of fields
3487:
3429:
3229:
3223:
3172:
3166:
3051:
3007:
3006:{\displaystyle p(x)=x}
2972:
2944:
2900:
2856:
2795:
2724:
2653:
2620:
2587:
2539:
2487:
2448:
2419:
2390:
2341:
2307:
2276:
2245:
2223:
2197:
2175:
2144:
2092:
2040:
2017:
1978:
1949:
1920:
1871:
1822:
1793:
1759:
1732:
1697:
1664:
1644:
1617:
1597:
1577:
1554:
1526:
1503:
1483:
1455:
1432:
1410:
1387:
1365:
1338:
1318:
1291:
1271:
1245:
1225:
1203:
1176:
1155:Its non-zero elements
1149:
1115:
1092:
1067:
1025:
986:
950:
919:
896:
870:
844:
821:
801:
777:
748:
708:
707:{\displaystyle a>0}
682:
681:{\displaystyle a\in F}
656:
655:{\displaystyle a<b}
630:
604:
603:{\displaystyle b>a}
578:
552:
526:
500:
499:{\displaystyle a<b}
471:
436:
410:
381:
340:
311:
262:
242:
219:
153:
119:of ordered fields and
4140:Lam (1983) p. 45
4131:Lam (1983) p. 39
3836:
3706:is equipped with the
3488:
3409:
3394:An ordered field has
3224:
3178:
3167:
3115:
3059:formal Laurent series
3052:
3008:
2973:
2945:
2901:
2857:
2796:
2725:
2654:
2621:
2588:
2540:
2488:
2449:
2420:
2391:
2342:
2308:
2277:
2246:
2224:
2198:
2176:
2145:
2093:
2041:
2025:transcendental number
2018:
1979:
1950:
1921:
1872:
1823:
1794:
1760:
1733:
1698:
1665:
1645:
1618:
1598:
1578:
1555:
1527:
1504:
1484:
1456:
1433:
1411:
1388:
1366:
1339:
1319:
1292:
1272:
1246:
1226:
1204:
1177:
1175:{\displaystyle P^{*}}
1150:
1116:
1093:
1068:
1026:
987:
951:
920:
897:
871:
845:
822:
802:
778:
749:
714:are called positive.
709:
683:
657:
631:
605:
579:
553:
527:
501:
472:
437:
411:
382:
341:
312:
263:
243:
241:{\displaystyle \leq }
220:
154:
152:{\displaystyle \leq }
117:Artin–Schreier theory
4900:Ordered vector space
3953:Ordered vector space
3859:totally disconnected
3779:
3663: = 0, and
3405:
3183:
3120:
3024:
2982:
2954:
2910:
2866:
2805:
2734:
2663:
2630:
2597:
2549:
2497:
2462:
2447:{\displaystyle q(x)}
2429:
2418:{\displaystyle p(x)}
2400:
2354:
2320:
2286:
2255:
2233:
2207:
2185:
2154:
2102:
2050:
2030:
1992:
1977:{\displaystyle q(x)}
1959:
1948:{\displaystyle p(x)}
1930:
1884:
1850:
1810:
1781:
1742:
1738:This total ordering
1707:
1674:
1654:
1627:
1607:
1587:
1564:
1538:
1513:
1493:
1473:
1442:
1422:
1418:The preorderings on
1397:
1377:
1352:
1328:
1305:
1281:
1255:
1235:
1215:
1190:
1159:
1136:
1102:
1079:
1035:
996:
960:
931:
906:
880:
854:
831:
811:
791:
761:
738:
692:
666:
640:
614:
588:
562:
536:
510:
484:
446:
420:
394:
350:
324:
280:
252:
232:
190:
143:
121:formally real fields
4738:Alexandrov topology
4684:Lexicographic order
4643:Well-quasi-ordering
4045:University of Liège
3612:formally real field
3521:. For example, the
3444:
1270:{\displaystyle -P,}
869:{\displaystyle x+y}
480:As usual, we write
175:partial orderings.
96:cannot be ordered.
4719:Transitive closure
4679:Converse/Transpose
4388:Dilworth's theorem
3912:superordered field
3831:
3483:
3482:
3430:
3230:
3219:
3173:
3162:
3075:real closed fields
3047:
3003:
2968:
2940:
2896:
2852:
2791:
2720:
2649:
2616:
2583:
2535:
2483:
2444:
2415:
2386:
2349:rational functions
2337:
2303:
2272:
2241:
2219:
2193:
2171:
2140:
2088:
2036:
2013:
1974:
1945:
1916:
1879:rational functions
1867:
1841:computable numbers
1818:
1789:
1755:
1728:
1693:
1660:
1640:
1613:
1593:
1576:{\displaystyle F.}
1573:
1550:
1525:{\displaystyle F.}
1522:
1499:
1479:
1454:{\displaystyle F.}
1451:
1428:
1409:{\displaystyle P.}
1406:
1383:
1364:{\displaystyle F.}
1361:
1334:
1317:{\displaystyle F.}
1314:
1287:
1267:
1241:
1221:
1202:{\displaystyle F.}
1199:
1172:
1148:{\displaystyle P.}
1145:
1114:{\displaystyle P.}
1111:
1091:{\displaystyle -1}
1088:
1063:
1021:
982:
946:
918:{\displaystyle P.}
915:
892:
866:
843:{\displaystyle P,}
840:
817:
797:
773:
744:
704:
678:
652:
626:
600:
574:
548:
522:
496:
467:
432:
406:
377:
336:
307:
258:
238:
215:
149:
99:Historically, the
4947:
4946:
4905:Partially ordered
4714:Symmetric closure
4699:Reflexive closure
4442:
4239:978-0-201-55540-0
4113:Lam (2005) p. 271
4063:Lam (2005) p. 236
4026:Lam (2005) p. 232
4002:Lam (2005) p. 289
3886:is a preordering
3761:subspace topology
3753:discrete topology
3732:Harrison topology
3726:Harrison topology
3720:topological field
3558:least upper bound
3540:An ordered field
3527:hyperreal numbers
3087:hyperreal numbers
3081:superreal numbers
1837:algebraic numbers
1663:{\displaystyle F}
1616:{\displaystyle F}
1596:{\displaystyle P}
1502:{\displaystyle F}
1482:{\displaystyle F}
1431:{\displaystyle F}
1386:{\displaystyle F}
1337:{\displaystyle P}
1290:{\displaystyle P}
1244:{\displaystyle P}
1224:{\displaystyle F}
820:{\displaystyle y}
800:{\displaystyle x}
747:{\displaystyle F}
261:{\displaystyle F}
71:Dedekind-complete
16:(Redirected from
4977:
4689:Linear extension
4438:
4418:Mirsky's theorem
4278:
4271:
4264:
4255:
4250:
4221:
4187:
4164:
4141:
4138:
4132:
4129:
4123:
4120:
4114:
4111:
4105:
4103:
4088:
4087:
4078:
4077:
4070:
4064:
4061:
4055:
4054:
4052:
4051:
4042:
4033:
4027:
4024:
4015:
4014:Lam (2005) p. 41
4012:
4003:
4000:
3976:
3944:
3935:
3840:
3838:
3837:
3832:
3815:
3814:
3757:product topology
3694: = 0.
3693:
3692:
3662:
3661:
3492:
3490:
3489:
3484:
3475:
3474:
3443:
3438:
3428:
3423:
3341:and c ≤ 0, then
3228:
3226:
3225:
3220:
3171:
3169:
3168:
3163:
3056:
3054:
3053:
3048:
3031:
3012:
3010:
3009:
3004:
2977:
2975:
2974:
2969:
2967:
2949:
2947:
2946:
2941:
2905:
2903:
2902:
2897:
2861:
2859:
2858:
2853:
2842:
2800:
2798:
2797:
2792:
2790:
2789:
2771:
2770:
2761:
2760:
2729:
2727:
2726:
2721:
2719:
2718:
2700:
2699:
2690:
2689:
2658:
2656:
2655:
2650:
2642:
2641:
2625:
2623:
2622:
2617:
2609:
2608:
2592:
2590:
2589:
2584:
2576:
2575:
2566:
2561:
2560:
2544:
2542:
2541:
2536:
2516:
2492:
2490:
2489:
2484:
2453:
2451:
2450:
2445:
2424:
2422:
2421:
2416:
2395:
2393:
2392:
2387:
2373:
2346:
2344:
2343:
2338:
2327:
2312:
2310:
2309:
2304:
2293:
2281:
2279:
2278:
2273:
2262:
2250:
2248:
2247:
2242:
2240:
2228:
2226:
2225:
2220:
2202:
2200:
2199:
2194:
2192:
2180:
2178:
2177:
2172:
2161:
2149:
2147:
2146:
2141:
2121:
2097:
2095:
2094:
2089:
2069:
2045:
2043:
2042:
2037:
2022:
2020:
2019:
2014:
1983:
1981:
1980:
1975:
1954:
1952:
1951:
1946:
1925:
1923:
1922:
1917:
1903:
1876:
1874:
1873:
1868:
1857:
1827:
1825:
1824:
1819:
1817:
1801:rational numbers
1798:
1796:
1795:
1790:
1788:
1764:
1762:
1761:
1756:
1754:
1753:
1737:
1735:
1734:
1729:
1702:
1700:
1699:
1694:
1689:
1688:
1669:
1667:
1666:
1661:
1649:
1647:
1646:
1641:
1639:
1638:
1622:
1620:
1619:
1614:
1602:
1600:
1599:
1594:
1582:
1580:
1579:
1574:
1559:
1557:
1556:
1551:
1531:
1529:
1528:
1523:
1508:
1506:
1505:
1500:
1488:
1486:
1485:
1480:
1460:
1458:
1457:
1452:
1437:
1435:
1434:
1429:
1415:
1413:
1412:
1407:
1392:
1390:
1389:
1384:
1370:
1368:
1367:
1362:
1343:
1341:
1340:
1335:
1323:
1321:
1320:
1315:
1296:
1294:
1293:
1288:
1276:
1274:
1273:
1268:
1250:
1248:
1247:
1242:
1231:is the union of
1230:
1228:
1227:
1222:
1208:
1206:
1205:
1200:
1181:
1179:
1178:
1173:
1171:
1170:
1154:
1152:
1151:
1146:
1130:
1129:
1128:preordered field
1120:
1118:
1117:
1112:
1097:
1095:
1094:
1089:
1072:
1070:
1069:
1064:
1053:
1052:
1030:
1028:
1027:
1022:
1014:
1013:
991:
989:
988:
983:
972:
971:
955:
953:
952:
947:
924:
922:
921:
916:
901:
899:
898:
893:
875:
873:
872:
867:
849:
847:
846:
841:
826:
824:
823:
818:
806:
804:
803:
798:
782:
780:
779:
774:
753:
751:
750:
745:
728:
727:
726:prepositive cone
713:
711:
710:
705:
687:
685:
684:
679:
661:
659:
658:
653:
635:
633:
632:
627:
609:
607:
606:
601:
583:
581:
580:
575:
558:. The notations
557:
555:
554:
549:
531:
529:
528:
523:
505:
503:
502:
497:
476:
474:
473:
468:
441:
439:
438:
433:
415:
413:
412:
407:
386:
384:
383:
378:
345:
343:
342:
337:
316:
314:
313:
308:
274:
273:
267:
265:
264:
259:
247:
245:
244:
239:
225:together with a
224:
222:
221:
216:
161:binary predicate
158:
156:
155:
150:
67:rational numbers
48:rational numbers
42:together with a
21:
18:Preordered field
4985:
4984:
4980:
4979:
4978:
4976:
4975:
4974:
4950:
4949:
4948:
4943:
4939:Young's lattice
4795:
4723:
4662:
4512:Heyting algebra
4460:Boolean algebra
4432:
4413:Laver's theorem
4361:
4327:Boolean algebra
4322:Binary relation
4310:
4287:
4282:
4240:
4224:
4210:
4190:
4177:
4153:
4150:
4145:
4144:
4139:
4135:
4130:
4126:
4121:
4117:
4112:
4108:
4098:
4082:
4080:
4075:
4073:
4071:
4067:
4062:
4058:
4049:
4047:
4040:
4035:
4034:
4030:
4025:
4018:
4013:
4006:
4001:
3994:
3989:
3974:
3942:
3933:
3920:
3876:
3869:
3806:
3777:
3776:
3771:
3742:
3728:
3700:
3687:
3685:
3671:
3659:
3657:
3655:
3633:complex numbers
3608:
3575:(particularly,
3570:
3466:
3403:
3402:
3368:> 0, then 1/
3181:
3180:
3118:
3117:
3110:
3094:surreal numbers
3022:
3021:
2980:
2979:
2952:
2951:
2908:
2907:
2906:if and only if
2864:
2863:
2803:
2802:
2781:
2762:
2752:
2732:
2731:
2710:
2691:
2681:
2661:
2660:
2633:
2628:
2627:
2600:
2595:
2594:
2567:
2552:
2547:
2546:
2495:
2494:
2460:
2459:
2427:
2426:
2398:
2397:
2352:
2351:
2318:
2317:
2284:
2283:
2253:
2252:
2231:
2230:
2205:
2204:
2183:
2182:
2152:
2151:
2100:
2099:
2098:if and only if
2048:
2047:
2028:
2027:
1990:
1989:
1957:
1956:
1928:
1927:
1882:
1881:
1848:
1847:
1808:
1807:
1779:
1778:
1771:
1745:
1740:
1739:
1705:
1704:
1680:
1672:
1671:
1652:
1651:
1630:
1625:
1624:
1605:
1604:
1585:
1584:
1562:
1561:
1536:
1535:
1511:
1510:
1491:
1490:
1471:
1470:
1467:
1440:
1439:
1420:
1419:
1395:
1394:
1375:
1374:
1350:
1349:
1344:are called the
1326:
1325:
1303:
1302:
1279:
1278:
1253:
1252:
1233:
1232:
1213:
1212:
1188:
1187:
1162:
1157:
1156:
1134:
1133:
1127:
1126:
1100:
1099:
1077:
1076:
1044:
1033:
1032:
1005:
994:
993:
992:In particular,
963:
958:
957:
929:
928:
904:
903:
878:
877:
852:
851:
829:
828:
809:
808:
789:
788:
759:
758:
736:
735:
725:
724:
720:
690:
689:
664:
663:
638:
637:
612:
611:
586:
585:
560:
559:
534:
533:
508:
507:
482:
481:
444:
443:
418:
417:
392:
391:
348:
347:
322:
321:
278:
277:
271:
270:
250:
249:
230:
229:
188:
187:
181:
141:
140:
129:
79:complex numbers
28:
23:
22:
15:
12:
11:
5:
4983:
4981:
4973:
4972:
4970:Ordered groups
4967:
4962:
4952:
4951:
4945:
4944:
4942:
4941:
4936:
4931:
4930:
4929:
4919:
4918:
4917:
4912:
4907:
4897:
4896:
4895:
4885:
4880:
4879:
4878:
4873:
4866:Order morphism
4863:
4862:
4861:
4851:
4846:
4841:
4836:
4831:
4830:
4829:
4819:
4814:
4809:
4803:
4801:
4797:
4796:
4794:
4793:
4792:
4791:
4786:
4784:Locally convex
4781:
4776:
4766:
4764:Order topology
4761:
4760:
4759:
4757:Order topology
4754:
4744:
4734:
4732:
4725:
4724:
4722:
4721:
4716:
4711:
4706:
4701:
4696:
4691:
4686:
4681:
4676:
4670:
4668:
4664:
4663:
4661:
4660:
4650:
4640:
4635:
4630:
4625:
4620:
4615:
4610:
4605:
4604:
4603:
4593:
4588:
4587:
4586:
4581:
4576:
4571:
4569:Chain-complete
4561:
4556:
4555:
4554:
4549:
4544:
4539:
4534:
4524:
4519:
4514:
4509:
4504:
4494:
4489:
4484:
4479:
4474:
4469:
4468:
4467:
4457:
4452:
4446:
4444:
4434:
4433:
4431:
4430:
4425:
4420:
4415:
4410:
4405:
4400:
4395:
4390:
4385:
4380:
4375:
4369:
4367:
4363:
4362:
4360:
4359:
4354:
4349:
4344:
4339:
4334:
4329:
4324:
4318:
4316:
4312:
4311:
4309:
4308:
4303:
4298:
4292:
4289:
4288:
4283:
4281:
4280:
4273:
4266:
4258:
4252:
4251:
4238:
4222:
4208:
4192:Lam, Tsit-Yuen
4188:
4175:
4149:
4146:
4143:
4142:
4133:
4124:
4115:
4106:
4083:1 −
4065:
4056:
4028:
4016:
4004:
3991:
3990:
3988:
3985:
3984:
3983:
3977:
3971:Preorder field
3968:
3962:
3956:
3950:
3945:
3936:
3927:
3919:
3916:
3875:
3872:
3865:
3830:
3827:
3824:
3821:
3818:
3813:
3809:
3805:
3802:
3799:
3796:
3793:
3790:
3787:
3784:
3767:
3738:
3727:
3724:
3708:order topology
3699:
3696:
3688:1 −
3667:
3653:
3648:Hensel's lemma
3629:characteristic
3607:
3604:
3569:
3566:
3535:natural number
3519:infinitesimals
3494:
3493:
3481:
3478:
3473:
3469:
3465:
3461:
3458:
3454:
3450:
3447:
3442:
3437:
3433:
3427:
3422:
3419:
3416:
3412:
3399:
3396:characteristic
3392:
3377:
3350:
3331:
3308:
3273:
3218:
3215:
3212:
3209:
3206:
3203:
3200:
3197:
3194:
3191:
3188:
3161:
3158:
3155:
3152:
3149:
3146:
3143:
3140:
3137:
3134:
3131:
3128:
3125:
3109:
3106:
3100:rather than a
3090:
3089:
3083:
3077:
3072:
3066:
3046:
3043:
3040:
3037:
3034:
3030:
3018:
3002:
2999:
2996:
2993:
2990:
2987:
2966:
2962:
2959:
2939:
2936:
2933:
2930:
2927:
2924:
2921:
2918:
2915:
2895:
2892:
2889:
2886:
2883:
2880:
2877:
2874:
2871:
2851:
2848:
2845:
2841:
2837:
2834:
2831:
2828:
2825:
2822:
2819:
2816:
2813:
2810:
2788:
2784:
2780:
2777:
2774:
2769:
2765:
2759:
2755:
2751:
2748:
2745:
2742:
2739:
2717:
2713:
2709:
2706:
2703:
2698:
2694:
2688:
2684:
2680:
2677:
2674:
2671:
2668:
2648:
2645:
2640:
2636:
2615:
2612:
2607:
2603:
2582:
2579:
2574:
2570:
2565:
2559:
2555:
2534:
2531:
2528:
2525:
2522:
2519:
2515:
2511:
2508:
2505:
2502:
2482:
2479:
2476:
2473:
2470:
2467:
2443:
2440:
2437:
2434:
2414:
2411:
2408:
2405:
2385:
2382:
2379:
2376:
2372:
2368:
2365:
2362:
2359:
2336:
2333:
2330:
2326:
2314:
2302:
2299:
2296:
2292:
2271:
2268:
2265:
2261:
2239:
2218:
2215:
2212:
2191:
2170:
2167:
2164:
2160:
2139:
2136:
2133:
2130:
2127:
2124:
2120:
2116:
2113:
2110:
2107:
2087:
2084:
2081:
2078:
2075:
2072:
2068:
2064:
2061:
2058:
2055:
2035:
2012:
2009:
2006:
2003:
2000:
1997:
1973:
1970:
1967:
1964:
1944:
1941:
1938:
1935:
1915:
1912:
1909:
1906:
1902:
1898:
1895:
1892:
1889:
1866:
1863:
1860:
1856:
1844:
1833:
1816:
1804:
1787:
1770:
1767:
1752:
1748:
1727:
1724:
1721:
1718:
1715:
1712:
1692:
1687:
1683:
1679:
1659:
1637:
1633:
1612:
1592:
1572:
1569:
1549:
1546:
1543:
1521:
1518:
1498:
1478:
1466:
1463:
1450:
1447:
1427:
1405:
1402:
1382:
1360:
1357:
1333:
1313:
1310:
1286:
1266:
1263:
1260:
1240:
1220:
1198:
1195:
1169:
1165:
1144:
1141:
1122:
1121:
1110:
1107:
1087:
1084:
1073:
1062:
1059:
1056:
1051:
1047:
1043:
1040:
1020:
1017:
1012:
1008:
1004:
1001:
981:
978:
975:
970:
966:
945:
942:
939:
936:
925:
914:
911:
891:
888:
885:
865:
862:
859:
839:
836:
816:
796:
772:
769:
766:
743:
719:
716:
703:
700:
697:
677:
674:
671:
651:
648:
645:
625:
622:
619:
599:
596:
593:
573:
570:
567:
547:
544:
541:
521:
518:
515:
495:
492:
489:
478:
477:
466:
463:
460:
457:
454:
451:
431:
428:
425:
405:
402:
399:
388:
376:
373:
370:
367:
364:
361:
358:
355:
335:
332:
329:
306:
303:
300:
297:
294:
291:
288:
285:
257:
237:
214:
210:
207:
204:
201:
198:
195:
180:
177:
174:
170:
148:
128:
125:
101:axiomatization
83:imaginary unit
44:total ordering
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4982:
4971:
4968:
4966:
4963:
4961:
4958:
4957:
4955:
4940:
4937:
4935:
4932:
4928:
4925:
4924:
4923:
4920:
4916:
4913:
4911:
4908:
4906:
4903:
4902:
4901:
4898:
4894:
4891:
4890:
4889:
4888:Ordered field
4886:
4884:
4881:
4877:
4874:
4872:
4869:
4868:
4867:
4864:
4860:
4857:
4856:
4855:
4852:
4850:
4847:
4845:
4844:Hasse diagram
4842:
4840:
4837:
4835:
4832:
4828:
4825:
4824:
4823:
4822:Comparability
4820:
4818:
4815:
4813:
4810:
4808:
4805:
4804:
4802:
4798:
4790:
4787:
4785:
4782:
4780:
4777:
4775:
4772:
4771:
4770:
4767:
4765:
4762:
4758:
4755:
4753:
4750:
4749:
4748:
4745:
4743:
4739:
4736:
4735:
4733:
4730:
4726:
4720:
4717:
4715:
4712:
4710:
4707:
4705:
4702:
4700:
4697:
4695:
4694:Product order
4692:
4690:
4687:
4685:
4682:
4680:
4677:
4675:
4672:
4671:
4669:
4667:Constructions
4665:
4659:
4655:
4651:
4648:
4644:
4641:
4639:
4636:
4634:
4631:
4629:
4626:
4624:
4621:
4619:
4616:
4614:
4611:
4609:
4606:
4602:
4599:
4598:
4597:
4594:
4592:
4589:
4585:
4582:
4580:
4577:
4575:
4572:
4570:
4567:
4566:
4565:
4564:Partial order
4562:
4560:
4557:
4553:
4552:Join and meet
4550:
4548:
4545:
4543:
4540:
4538:
4535:
4533:
4530:
4529:
4528:
4525:
4523:
4520:
4518:
4515:
4513:
4510:
4508:
4505:
4503:
4499:
4495:
4493:
4490:
4488:
4485:
4483:
4480:
4478:
4475:
4473:
4470:
4466:
4463:
4462:
4461:
4458:
4456:
4453:
4451:
4450:Antisymmetric
4448:
4447:
4445:
4441:
4435:
4429:
4426:
4424:
4421:
4419:
4416:
4414:
4411:
4409:
4406:
4404:
4401:
4399:
4396:
4394:
4391:
4389:
4386:
4384:
4381:
4379:
4376:
4374:
4371:
4370:
4368:
4364:
4358:
4357:Weak ordering
4355:
4353:
4350:
4348:
4345:
4343:
4342:Partial order
4340:
4338:
4335:
4333:
4330:
4328:
4325:
4323:
4320:
4319:
4317:
4313:
4307:
4304:
4302:
4299:
4297:
4294:
4293:
4290:
4286:
4279:
4274:
4272:
4267:
4265:
4260:
4259:
4256:
4249:
4245:
4241:
4235:
4231:
4227:
4223:
4219:
4215:
4211:
4209:0-8218-1095-2
4205:
4201:
4197:
4193:
4189:
4186:
4182:
4178:
4176:0-8218-0702-1
4172:
4168:
4163:
4162:
4156:
4152:
4151:
4147:
4137:
4134:
4128:
4125:
4119:
4116:
4110:
4107:
4101:
4096:
4092:
4086:
4069:
4066:
4060:
4057:
4046:
4039:
4032:
4029:
4023:
4021:
4017:
4011:
4009:
4005:
3999:
3997:
3993:
3986:
3981:
3978:
3972:
3969:
3966:
3963:
3960:
3957:
3954:
3951:
3949:
3946:
3940:
3937:
3931:
3930:Ordered group
3928:
3925:
3922:
3921:
3917:
3915:
3913:
3909:
3905:
3901:
3897:
3893:
3889:
3885:
3881:
3873:
3871:
3868:
3864:
3860:
3856:
3852:
3848:
3847:Boolean space
3844:
3825:
3822:
3819:
3816:
3811:
3807:
3803:
3800:
3794:
3788:
3782:
3775:
3774:Harrison sets
3770:
3766:
3762:
3758:
3754:
3750:
3746:
3741:
3737:
3733:
3725:
3723:
3721:
3717:
3713:
3709:
3705:
3697:
3695:
3691:
3683:
3679:
3675:
3670:
3666:
3652:
3649:
3645:
3644:-adic numbers
3643:
3638:
3634:
3630:
3626:
3625:Finite fields
3622:
3620:
3615:
3613:
3605:
3603:
3601:
3597:
3593:
3589:
3585:
3581:
3579:
3574:
3573:Vector spaces
3567:
3565:
3563:
3559:
3555:
3551:
3547:
3543:
3538:
3536:
3532:
3528:
3524:
3520:
3517:and contains
3516:
3512:
3511:
3505:
3503:
3499:
3479:
3476:
3471:
3467:
3463:
3459:
3448:
3445:
3440:
3435:
3431:
3425:
3420:
3417:
3414:
3410:
3400:
3397:
3393:
3390:
3386:
3382:
3378:
3375:
3371:
3367:
3363:
3359:
3355:
3351:
3348:
3344:
3340:
3336:
3332:
3329:
3325:
3321:
3317:
3313:
3309:
3306:
3302:
3298:
3294:
3290:
3286:
3282:
3278:
3274:
3271:
3267:
3263:
3259:
3255:
3254:
3253:
3251:
3247:
3243:
3239:
3235:
3216:
3213:
3210:
3207:
3204:
3201:
3198:
3192:
3189:
3186:
3179:The property
3177:
3159:
3156:
3153:
3150:
3147:
3141:
3138:
3135:
3132:
3129:
3126:
3123:
3116:The property
3114:
3107:
3105:
3103:
3099:
3095:
3088:
3084:
3082:
3078:
3076:
3073:
3071:
3067:
3064:
3060:
3038:
3019:
3016:
3000:
2997:
2991:
2985:
2960:
2957:
2934:
2928:
2925:
2919:
2913:
2890:
2884:
2881:
2875:
2869:
2846:
2835:
2829:
2823:
2820:
2814:
2808:
2786:
2782:
2778:
2775:
2772:
2767:
2763:
2757:
2753:
2749:
2743:
2737:
2715:
2711:
2707:
2704:
2701:
2696:
2692:
2686:
2682:
2678:
2672:
2666:
2646:
2643:
2638:
2634:
2613:
2610:
2605:
2601:
2580:
2577:
2572:
2568:
2563:
2557:
2553:
2545:to mean that
2532:
2529:
2523:
2517:
2513:
2506:
2500:
2480:
2477:
2471:
2465:
2457:
2438:
2432:
2409:
2403:
2380:
2374:
2370:
2363:
2357:
2350:
2331:
2315:
2297:
2266:
2216:
2210:
2165:
2137:
2134:
2128:
2122:
2118:
2111:
2105:
2085:
2082:
2076:
2070:
2066:
2059:
2053:
2046:and defining
2033:
2026:
2010:
2007:
2001:
1995:
1987:
1968:
1962:
1939:
1933:
1910:
1904:
1900:
1893:
1887:
1880:
1861:
1845:
1842:
1838:
1834:
1831:
1805:
1802:
1776:
1775:
1774:
1768:
1766:
1750:
1746:
1725:
1722:
1719:
1716:
1713:
1710:
1690:
1685:
1681:
1677:
1657:
1635:
1631:
1610:
1590:
1570:
1567:
1547:
1544:
1541:
1532:
1519:
1516:
1496:
1476:
1464:
1462:
1448:
1445:
1425:
1416:
1403:
1400:
1380:
1371:
1358:
1355:
1347:
1331:
1311:
1308:
1300:
1299:positive cone
1284:
1264:
1261:
1258:
1238:
1218:
1209:
1196:
1193:
1185:
1167:
1163:
1142:
1139:
1131:
1108:
1105:
1085:
1082:
1074:
1060:
1057:
1054:
1049:
1045:
1041:
1038:
1018:
1015:
1010:
1006:
1002:
999:
979:
976:
973:
968:
964:
943:
940:
937:
934:
926:
912:
909:
889:
886:
883:
863:
860:
857:
837:
834:
814:
794:
786:
785:
784:
770:
767:
764:
757:
741:
733:
729:
718:Positive cone
717:
715:
701:
698:
695:
675:
672:
669:
649:
646:
643:
623:
620:
617:
597:
594:
591:
571:
568:
565:
545:
542:
539:
519:
516:
513:
493:
490:
487:
464:
461:
458:
455:
452:
449:
429:
426:
423:
403:
400:
397:
389:
374:
371:
368:
365:
362:
359:
356:
353:
333:
330:
327:
319:
318:
317:
304:
301:
298:
295:
292:
289:
286:
283:
275:
272:ordered field
255:
235:
228:
208:
205:
202:
199:
196:
186:
178:
176:
172:
168:
166:
165:positive cone
162:
146:
138:
134:
126:
124:
122:
118:
114:
110:
106:
105:David Hilbert
102:
97:
95:
94:Finite fields
91:
87:
84:
80:
76:
72:
68:
64:
60:
55:
53:
49:
45:
41:
37:
36:ordered field
33:
19:
4887:
4731:& Orders
4709:Star product
4638:Well-founded
4591:Prefix order
4547:Distributive
4537:Complemented
4507:Foundational
4472:Completeness
4428:Zorn's lemma
4332:Cyclic order
4315:Key concepts
4285:Order theory
4229:
4195:
4160:
4136:
4127:
4118:
4109:
4099:
4094:
4090:
4084:
4068:
4059:
4048:. Retrieved
4031:
3939:Ordered ring
3911:
3907:
3903:
3899:
3895:
3891:
3887:
3883:
3879:
3877:
3866:
3862:
3773:
3768:
3764:
3759:induces the
3748:
3744:
3739:
3735:
3731:
3729:
3715:
3703:
3701:
3689:
3681:
3677:
3673:
3668:
3664:
3650:
3641:
3639:. Also, the
3636:
3623:
3619:Zorn's lemma
3616:
3609:
3599:
3577:
3571:
3561:
3553:
3549:
3545:
3541:
3539:
3523:real numbers
3508:
3506:
3495:
3388:
3384:
3380:
3373:
3369:
3365:
3361:
3357:
3353:
3346:
3342:
3338:
3334:
3327:
3323:
3319:
3315:
3311:
3304:
3300:
3296:
3292:
3288:
3284:
3280:
3276:
3269:
3265:
3261:
3257:
3249:
3245:
3241:
3237:
3233:
3231:
3098:proper class
3091:
3062:
1830:real numbers
1772:
1533:
1468:
1417:
1372:
1348:elements of
1345:
1298:
1210:
1125:
1123:
1075:The element
731:
723:
721:
479:
269:
182:
164:
132:
130:
98:
85:
56:
52:real numbers
35:
29:
4915:Riesz space
4876:Isomorphism
4752:Normal cone
4674:Composition
4608:Semilattice
4517:Homogeneous
4502:Equivalence
4352:Total order
4226:Lang, Serge
3980:Riesz space
3898:containing
3755:and ±1 the
3584:orientation
3510:Archimedean
3070:transseries
3015:Archimedean
2456:polynomials
1986:polynomials
1670:by setting
734:of a field
732:preordering
227:total order
179:Total order
137:first-order
133:total order
127:Definitions
109:Otto Hölder
32:mathematics
4954:Categories
4883:Order type
4817:Cofinality
4658:Well-order
4633:Transitive
4522:Idempotent
4455:Asymmetric
4248:0848.13001
4218:1068.11023
4185:0516.12001
4155:Lam, T. Y.
4148:References
4050:2013-05-04
3714:, so that
3712:continuous
3498:isomorphic
3232:For every
3020:The field
2316:the field
1846:the field
1806:the field
1777:the field
1098:is not in
610:stand for
63:isomorphic
4934:Upper set
4871:Embedding
4807:Antichain
4628:Tolerance
4618:Symmetric
4613:Semiorder
4559:Reflexive
4477:Connected
3855:Hausdorff
3823:∈
3804:∈
3588:convexity
3502:rationals
3464::
3457:∀
3453:⟹
3411:∑
3196:⇒
3145:⇒
3133:∧
2961:∈
2836:∈
2776:⋯
2705:⋯
2644:≠
2611:≠
2478:≠
2217:α
2214:↦
2129:α
2112:α
2034:α
2008:≠
1747:≤
1720:∈
1714:−
1682:≤
1632:≤
1545:≥
1259:−
1168:∗
1083:−
1055:∈
1016:∈
974:∈
938:∈
887:⋅
768:⊆
673:∈
621:≤
569:≥
543:≠
517:≤
459:⋅
453:≤
427:≤
401:≤
363:≤
331:≤
299:∈
236:≤
209:⋅
147:≤
113:Hans Hahn
4729:Topology
4596:Preorder
4579:Eulerian
4542:Complete
4492:Directed
4482:Covering
4347:Preorder
4306:Category
4301:Glossary
4228:(1993),
4194:(2005).
4157:(1983),
3918:See also
3843:subbasis
3680:, thus (
3560:in
3383:for all
3318:and 0 ≤
3256:Either −
2862:we have
2593:, where
2396:, where
1926:, where
1703:to mean
1346:positive
1277:we call
1184:subgroup
173:extremal
69:. Every
59:subfield
50:and the
4834:Duality
4812:Cofinal
4800:Related
4779:Fréchet
4656:)
4532:Bounded
4527:Lattice
4500:)
4498:Partial
4366:Results
4337:Lattice
4230:Algebra
4089:are in
4081:√
4074:√
3861:), and
3851:compact
3841:form a
3772:. The
3686:√
3658:√
3580:-spaces
3531:extends
3500:to the
3372:< 1/
3322:, then
3291:, then
3268:≤ 0 ≤ −
3096:form a
1839:or the
1182:form a
902:are in
169:maximal
75:Squares
65:to the
4859:Subnet
4839:Filter
4789:Normed
4774:Banach
4740:&
4647:Better
4584:Strict
4574:Graded
4465:topics
4296:Topics
4246:
4236:
4216:
4206:
4183:
4173:
3594:. See
3590:, and
3556:has a
3260:≤ 0 ≤
756:subset
268:is an
57:Every
4849:Ideal
4827:Graph
4623:Total
4601:Total
4487:Dense
4102:-adic
4041:(PDF)
3987:Notes
3718:is a
3356:<
2181:into
1031:and
956:then
850:both
754:is a
688:with
442:then
346:then
185:field
159:as a
40:field
38:is a
34:, an
4440:list
4234:ISBN
4204:ISBN
4171:ISBN
4079:and
3857:and
3730:The
3360:and
3283:and
3208:<
3190:<
3154:<
3139:<
3127:>
3092:The
3085:the
3079:the
3068:the
2926:<
2882:<
2730:and
2626:and
2578:>
2530:>
2454:are
2425:and
2203:via
2135:>
2083:>
1984:are
1955:and
1469:Let
1251:and
876:and
807:and
787:For
699:>
647:<
636:and
595:>
584:and
532:and
506:for
491:<
416:and
111:and
4854:Net
4654:Pre
4244:Zbl
4214:Zbl
4181:Zbl
3882:on
3880:fan
3763:on
3702:If
3387:in
3352:If
3264:or
3248:in
3102:set
3057:of
2347:of
1877:of
1828:of
1799:of
1650:on
1603:of
1301:of
927:If
827:in
730:or
390:if
387:and
320:if
248:on
88:is
30:In
4956::
4242:,
4212:.
4198:.
4179:,
4169:,
4076:−7
4043:.
4019:^
4007:^
3995:^
3878:A
3853:,
3722:.
3660:−7
3621:.
3586:,
3537:.
3480:0.
3364:,
3347:bc
3345:≥
3343:ac
3337:≤
3328:bc
3326:≤
3324:ac
3314:≤
3303:+
3299:≤
3295:+
3287:≤
3279:≤
3252::
3244:,
3240:,
3236:,
1297:a
1124:A
722:A
183:A
123:.
107:,
90:−1
4652:(
4649:)
4645:(
4496:(
4443:)
4277:e
4270:t
4263:v
4220:.
4100:p
4095:Q
4091:Q
4085:p
4053:.
3908:S
3904:S
3900:T
3896:F
3892:S
3888:T
3884:F
3867:F
3863:X
3849:(
3829:}
3826:P
3820:a
3817::
3812:F
3808:X
3801:P
3798:{
3795:=
3792:)
3789:a
3786:(
3783:H
3769:F
3765:X
3749:F
3745:F
3740:F
3736:X
3716:F
3704:F
3690:p
3682:p
3678:p
3674:p
3672:(
3669:p
3665:Q
3654:2
3651:Q
3642:p
3637:i
3600:R
3578:n
3562:F
3554:F
3550:F
3546:R
3542:F
3477:=
3472:k
3468:a
3460:k
3449:0
3446:=
3441:2
3436:k
3432:a
3426:n
3421:1
3418:=
3415:k
3389:F
3385:a
3381:a
3376:.
3374:a
3370:b
3366:b
3362:a
3358:b
3354:a
3349:.
3339:b
3335:a
3330:.
3320:c
3316:b
3312:a
3307:.
3305:d
3301:b
3297:c
3293:a
3289:d
3285:c
3281:b
3277:a
3272:.
3270:a
3266:a
3262:a
3258:a
3250:F
3246:d
3242:c
3238:b
3234:a
3217:y
3214:+
3211:a
3205:x
3202:+
3199:a
3193:y
3187:x
3160:y
3157:a
3151:x
3148:a
3142:y
3136:x
3130:0
3124:a
3063:x
3045:)
3042:)
3039:x
3036:(
3033:(
3029:R
3017:.
3001:x
2998:=
2995:)
2992:x
2989:(
2986:p
2965:R
2958:t
2938:)
2935:t
2932:(
2929:g
2923:)
2920:t
2917:(
2914:f
2894:)
2891:x
2888:(
2885:g
2879:)
2876:x
2873:(
2870:f
2850:)
2847:x
2844:(
2840:R
2833:)
2830:x
2827:(
2824:g
2821:,
2818:)
2815:x
2812:(
2809:f
2787:0
2783:q
2779:+
2773:+
2768:m
2764:x
2758:m
2754:q
2750:=
2747:)
2744:x
2741:(
2738:q
2716:0
2712:p
2708:+
2702:+
2697:n
2693:x
2687:n
2683:p
2679:=
2676:)
2673:x
2670:(
2667:p
2647:0
2639:m
2635:q
2614:0
2606:n
2602:p
2581:0
2573:m
2569:q
2564:/
2558:n
2554:p
2533:0
2527:)
2524:x
2521:(
2518:q
2514:/
2510:)
2507:x
2504:(
2501:p
2481:0
2475:)
2472:x
2469:(
2466:q
2442:)
2439:x
2436:(
2433:q
2413:)
2410:x
2407:(
2404:p
2384:)
2381:x
2378:(
2375:q
2371:/
2367:)
2364:x
2361:(
2358:p
2335:)
2332:x
2329:(
2325:R
2313:.
2301:)
2298:x
2295:(
2291:Q
2270:)
2267:x
2264:(
2260:Q
2238:R
2211:x
2190:R
2169:)
2166:x
2163:(
2159:Q
2138:0
2132:)
2126:(
2123:q
2119:/
2115:)
2109:(
2106:p
2086:0
2080:)
2077:x
2074:(
2071:q
2067:/
2063:)
2060:x
2057:(
2054:p
2011:0
2005:)
2002:x
1999:(
1996:q
1972:)
1969:x
1966:(
1963:q
1943:)
1940:x
1937:(
1934:p
1914:)
1911:x
1908:(
1905:q
1901:/
1897:)
1894:x
1891:(
1888:p
1865:)
1862:x
1859:(
1855:Q
1815:R
1786:Q
1751:P
1726:.
1723:P
1717:x
1711:y
1691:y
1686:P
1678:x
1658:F
1636:P
1611:F
1591:P
1571:.
1568:F
1548:0
1542:x
1520:.
1517:F
1497:F
1477:F
1449:.
1446:F
1426:F
1404:.
1401:P
1381:F
1359:.
1356:F
1332:P
1312:.
1309:F
1285:P
1265:,
1262:P
1239:P
1219:F
1197:.
1194:F
1164:P
1143:.
1140:P
1109:.
1106:P
1086:1
1061:.
1058:P
1050:2
1046:1
1042:=
1039:1
1019:P
1011:2
1007:0
1003:=
1000:0
980:.
977:P
969:2
965:x
944:,
941:F
935:x
913:.
910:P
890:y
884:x
864:y
861:+
858:x
838:,
835:P
815:y
795:x
771:F
765:P
742:F
702:0
696:a
676:F
670:a
650:b
644:a
624:b
618:a
598:a
592:b
572:a
566:b
546:b
540:a
520:b
514:a
494:b
488:a
465:.
462:b
456:a
450:0
430:b
424:0
404:a
398:0
375:,
372:c
369:+
366:b
360:c
357:+
354:a
334:b
328:a
305::
302:F
296:c
293:,
290:b
287:,
284:a
256:F
213:)
206:,
203:+
200:,
197:F
194:(
86:i
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.